Scalar Field theory in $κ$-Minkowski spacetime from twist

0.2 MB·English

by  Hyeong-Chan Kim

Scalar Field theory in κ-Minkowski spacetime from twist
Hyeong-Chan Kim∗
Center for Quantum Space Time, Sogang University, Seoul 121-742, Republic of Korea
Youngone Lee†
Department of Physics, Daejin University, Pocheon, 487-711, Republic of Korea
9 Chaiho Rim‡
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2 Department of Physics and Research Institute of Physics and Chemistry,
Chonbuk National University, Jeonju 561-756, Korea
n
a
J Jae Hyung Yee§

0
3
Department of Physics, Yonsei University, Seoul 120-749, Republic of Korea

]
h
t
-
p Using the twist deformation of U(igl(4,R)), the linear partof the diffeomorphism,
e
h wedefineascalarfunctionandconstructafreescalarfieldtheoryinfour-dimensional
[ κ-Minkowski spacetime. The action in momentum space turns out to differ only in

2 integration measure from the commutative theory.
v
9
PACS numbers: 02.20.Uw, 02.40.Gh
4
0 Keywords: κ-Minkowski spacetime, twist deformation, noncommutative field theory
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.
1
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0 I. INTRODUCTION
:
v
i Doplicher, Fredenhagen, and Roberts [1] showed that, in the presence of gravity, the
X
Heisenberg’s uncertainty relation has to be generalized to include the uncertainty between
r
a coordinates, which may be reproduced from the noncommutativity of coordinates such as
the canonical noncommutativity [2] or the κ-Minkowski spacetime [3, 4]. Especially in the
κ-Minkowski spacetime, the position xˆµ satisfies an algebra-like commutational relation,
i
[x0,xi] = xi, (1)
κ
with all other commutators vanishing.
The κ-Minkowski spacetime may arise as an effective low energy description of quantum
gravity [5, 6, 7]. Such space first appeared in the investigation of κ-Poincar´e algebra. Later,
∗Electronic address: [email protected]
†Electronic address: [email protected]
‡Electronic address: [email protected]
§Electronic address: [email protected]
2
it was related to Doubly Special Relativity (See [8] and references therein) which might have
a quantum gravitational origin [9].
The differential structure of the κ-Minkowski spacetime has been constructed in [10] and
based on this differential structure, the scalar field theory has been formulated [5, 11, 12, 13,
14, 15, 16]. It was shown that the differential structure requires that the momentum space
corresponding to the κ-Minkowski spacetime becomes a de-Sitter section in five-dimensional
flat space. Various physical aspects of κ-Minkowski spacetime have been investigated in
Refs. [16] and was extended to κ-Robertson-Walker spacetime [17]. The Fock space and
its symmetries [18, 19], κ-deformed statistics of particles [20, 21], and interpretation of the
κ-Minkowski spacetime in terms of exotic oscillator [22] were also studied. In addition, the
properties of scalar field theory on this spacetime has started being analyzed in depth [5,
11, 12, 13].
Recently, the κ-Minkowski spacetime is realized in terms of twisting procedure [23, 24,
25, 26, 27]. This twist approach can be seen as an alternative to the κ-like deformation of
the quantum Weyl and conformal algebra [27], which is obtained by using the Jordanian
twist [28, 29, 30]. The light-coneκ-deformationofPoincar´e algebra canbegivenby standard
twist (see eg. [23]). The realization for the time-like κ-deformation was constructed in
Ref. [24, 25] by embedding an abelian twist in igl(4,R) whose symmetry is bigger than
the Poincar´e, and their differential structure was studied in [31]. One can also find other
approach to the differential structure and twist realization of κ-Minkowski spacetime by
using the Weyl algebra [26].
In this paper, we construct a free scalar field theory by using the twist approach [24, 31].
In Sec. II, we review the κ-Minkowski spacetime from twist and then define the -product
∗
between vectors. We also provide an interesting relation between the generators when acting
on coordinates space. In Sec. III, we introduce a new action of the generators on function
and define a ⋆-product between functions. In Sec. IV, we find a transformation rule for
a scalar function and then construct an action for a real free scalar field in Sec. V. We
summarize the results and discuss the physical applications in Sec. VI.
II. REVIEW ON THE κ-MINKOWSKI SPACETIME FROM TWIST
Twisting the Hopf algebra of the universal enveloping algebra of inhomogeneous gen-
eral linear group is considered in [24, 25]. The group of inhomogeneous linear coordinate
transformations is composed of the product of the general linear transformations and the
spacetime translations. The inhomogeneous general linear algebra in (3+1)-dimensional flat
spacetime g = igl(4,R) is composed of 20 generators P ,Ma (a,b = 0,1,2,3) where P
{ a b} a
represents the spacetime translation and Ma represents the boost, rotations and dilations.
b
The generators satisfy the commutation relations,
[P ,P ] = 0, [Ma,P ] = iδa P ,
a b b c c · b
[Ma,Mc ] = i(δa Mc δc Ma ). (2)
b d d · b − b · d
The universal enveloping Hopf algebra (g, ,∆,ε,S) with the counit ε and antipode S can
U ·
be constructed starting from the base elements 1,P ,Ma and coproduct ∆Y = 1 Y +
{ a b} ⊗
Y 1 with Y P , Ma .
⊗ ∈ { a b}
The infinitesimal transformation by the general linear group is given in terms of igl(4,R)
3
generators:
δ S = iǫcY ⊲S, (3)
ǫ c
−
where ⊲ denotes an abstract action of Y on vectors, scalar fields, or vector fields. Consider
the action ⊲ of Y on an algebra = (V, ) where V xa k kc satisfying
b
V · ≡ { }∪{ }∪{ }
Ma ⊲xc = ixaδc, Ma ⊲k = ik δa, Ma ⊲kc = ikaδc, (4)
b − b b c b c b − a
P ⊲xb = iδb, P ⊲k = 0, P ⊲kb = 0.
a − a a b a
In the previous paper [31], we considered the vector space xa (e )b or equivalently
µ
{ } ∪ { }
xa kb . To define a scalar function in this paper we extend the module algebra of (g)
{ }∪{ } U
to include a dual space k to ka . Therefore, we generalize the relations in Eqs. (11)
a
{ } { }
and (12) in Ref. [31] to both of the covariant and contravariant vectors. The action on the
product of vectors is given by the coproduct,
Y ⊲(v v ) = (∆Y ⊲(v v )) = (Y ⊲v ) v +v (Y ⊲v ), (5)
1 2 1 2 1 2 1 2
· · ⊗ · ·
where v ,v . If we choose v = xa and v = xb, this equation provides the well-known
1 2 1 2
∈ V
Leibnitz rule.
1. Abelian twist
A new Hopf algebra is obtained by twisting a given Hopf algebra. The new Hopf algebra
has the algebra part in common with the original, however, the coproduct is changed by the
twist. A twist is a counital 2-cocycle satisfying (ε id) = 1 and (1 )(id ∆) =
κ κ κ κ
F ⊗ F ⊗F ⊗ F
( 1)(∆ id) . The new Hopf algebra, (g, ,∆ ,ε,S) is given by the original counit
κ κ κ κ
F ⊗ ⊗ F U ·
and antipode (ε = ε, S = S), but with a twisted coproduct:
κ κ
∆ (Y) = ∆Y −1 = Y Y Y Y . (6)
κ Fκ · ·Fκ (1)i ⊗ (2)i ≡ (1) ⊗ (2)
Xi
An abelian twist can be constructed by exponentiating two commuting generators such
as the momentum operators P and P , which gives the canonical noncommutativity [32].
1 2
Other choice of a twist by using two commuting operators E = P and D = 3 Mi [24]
0 i=1 i
P
i
= exp αE D (1 α)D E (7)
κ
F (cid:20)κ ⊗ − − ⊗ (cid:21)
(cid:16) (cid:17)
generates the κ-Minkowski spacetime with the twisted Hopf algebra (g). α is a constant
κ
U
representing different ordering of the exponential kernel function in the conventional κ-
Minkowski spacetime formulation. In this paper, α = 1/2, which corresponds to the time-
symmetric ordering.
For convenience, we explicitly write down the twisted coproduct, (i,j = 1,2,3)
∆ (Z) = Z 1+1 Z, Z E,D,Mi , (8)
κ ⊗ ⊗ ∈ { j}
∆ (P ) = P eE/(2κ) +e−E/(2κ) P ,
κ i i i
⊗ ⊗
∆ (Mi ) = Mi e−E/(2κ) +eE/(2κ) Mi ,
κ 0 0 ⊗ ⊗ 0
1
∆ (M0) = M0 eE/(2κ) +e−E/(2κ) M0 + P DeE/(2κ) e−E/(2κ)D P ,
κ i i ⊗ ⊗ i 2κ i ⊗ − ⊗ i
(cid:0) (cid:1)
1
∆ (M0 ) = M0 1+1 M0 + (E D D E).
κ 0 0 ⊗ ⊗ 0 2κ ⊗ − ⊗
4
The spatial (rotational) parts are undeformed and keeps the rotational symmetry. On the
other hand, the boost parts are deformed nontrivially due to the presence of the spatial
dilatation term. It is noted that the twisted Hopf algebra (g) is different from that of
κ
U
the conventional κ-Poincar´e in two aspects. First, the algebraic part is nothing but those of
the un-deformed inhomogeneous general linear group (2) rather than that of the deformed
Poincar´e. Second, the co-algebra structure is enlarged due to the bigger symmetry igl(4,R)
and its co-product is deformed as (8).
2. -product between vectors
∗
In this subsection, we study the non-commutative -product by using the twisted Hopf
∗
algebra. The twist (g) with new product, , given by
κ
F ∈ U ∗
v v [v v ] = −1 ⊲(v v ) , v ,v V, (9)
1 ∗ 2 ≡ ∗ 1 ⊗ 2 · Fκ 1 ⊗ 2 1 2 ∈
(cid:2) (cid:3)
defines a new associated algebra = (V, ) as a module algebra of (g) in the sense that
κ κ
V ∗ U
Y ⊲(v v ) = (∆ Y ⊲(v v )) = (Y ⊲v ) (Y ⊲v ). (10)
1 2 κ 1 2 (1) 1 (2) 2
∗ ∗ ⊗ ∗
X
Explicitly, the -product between xa and xb gives the usual κ-Minkowski relation (1) and the
∗
-product between xa and kb leads to the nontrivial commutation relation [x0,ki] = iki [31].
∗ κ
The -product with k gives
b
∗
i
k xb = k xb + δiδbk , (11)
a ∗ a 2κ a 0 i
i
xb k = k xb δiδbk ,
∗ a a − 2κ a 0 i
k q = k q ,k qb = k qb,
a b a b a a
∗ ∗
which results in commutation relations
i
[k ,q ] = 0, [x0,k ] = k , [x0,k ] = 0 = [xi,k ] . (12)
a b ∗ i ∗ i 0 ∗ a ∗
−κ
Note that any vector having non-vanishing spatial index does not commute with the time
coordinate. ItwasalsoshowninRef.[31]thattherelation(12)isrelatedtothe4-dimensional
differential structure of the κ-Minkowski spacetime from twist.
3. Relation between the actions of Ma and P
b c
In this subsection, we provide a nontrivial relation satisfied by the two actions of Ma
b
and P on the vector space of coordinate vector xa:
a
Ma ⊲(xc1 xc2 xcn) = xa[P ⊲(xc1 xc2 xcn)]. (13)
b ∗ ∗···∗ b ∗ ∗···∗
We prove this by the method of induction. We use the notation x(n) = (xc1 xc2 xcn)
∗ ∗···∗
for simplicity. In the case of x(1), it is clear from the definition of Ma in Eq. (4). Let us
b
5
assume that x(n) satisfies Eq. (13). Then, we can show that x(n+1) = xc0 x(n) also satisfies
∗
the relation. As an illustration we post the proof for the case of M0:
i
M0 ⊲xc0 x(n) = (M0 ⊲xc0) (eE/(2κ) ⊲x(n))+(e−E/(2κ) ⊲xc0) (M0 ⊲x(n)) (14)
i ∗ i ∗ ∗ i
1
+ (P ⊲xc0) (DeE/(2κ) ⊲x(n)) (De−E/(2κ) ⊲xc0) (P ⊲x(n))
i i
2κ ∗ − ∗
(cid:2) (cid:3)
where we use the deformed coproduct (8) and the definition (10) of the action on the -
∗
product. The first term in the right-hand-side of Eq. (14) becomes
i
iδc0 x0 (eE/(2κ) ⊲x(n)) = x0 iδc0 (eE/(2κ) ⊲x(n)) i(− )( iδc0) (DeE/(2κ) ⊲x(n))
− i ∗ − i ∗ − 2κ − i ∗
(cid:2) (cid:3)
1
= x0 (P ⊲xc0) (eE/(2κ) ⊲x(n)) (P ⊲xc0) (DeE/(2κ) ⊲x(n)),
i i
∗ − 2κ ∗
(cid:2) (cid:3)
where we use the definition of to replace the -product x0 [ ] with a normal one
∗ ∗ ∗ ···
x0[ ], by using x0 [ ] = x0[ ] 1 (D⊲[ ]) in the first equality and use the property
··· ∗ ··· ··· − 2κ ···
P ⊲x0 = iδx0 in the second equality. Similarly, the second term in the right-hand-side of
i − i
Eq. (14) becomes
(e−E/(2κ) ⊲xc0) [x0(P ⊲x(n))]
i
∗
1
= (e−E/(2κ) ⊲xc0) [x0 (P ⊲x(n))]+ (e−E/(2κ) ⊲xc0) (D ⊲(P ⊲x(n)))
i i
∗ ∗ 2κ ∗
= x0 (e−E/(2κ) ⊲xc0) (P ⊲x(n)) [x0,e−E/(2κ) ⊲xc0] (P ⊲x(n))
i ∗ i
∗ ∗ − ∗
1
+ (e−E/(2κ) ⊲xc0) (DP ⊲x(n))
i
2κ ∗
1
= x0 (e−E/(2κ) ⊲xc0) (P ⊲x(n)) + (De−E/(2κ) ⊲xc0) (P ⊲x(n))
i i
∗ 2κ ∗
(cid:2) (cid:3)
where we replace the normal product x0[ ] with a -product in the first equality, exchange
··· ∗
the order of product in the second equality, and then replace the -product x0 [ ] with
∗ ∗ ···
a normal product in the last equality. Adding the above two equations we have
(M0 ⊲xc0) (eE/(2κ) ⊲x(n))+(e−E/(2κ) ⊲xc0) (M0 ⊲x(n))
i ∗ ∗ i
1
= x0(P ⊲x(n+1)) (P ⊲xc0) (DeE/(2κ) ⊲x(n)) (De−E/(2κ) ⊲xc0) (P ⊲x(n)) .
i i i
− 2κ ∗ − ∗
(cid:2) (cid:3)
The two O(1/κ) terms exactly cancels the last two terms in Eq. (14). In the case of the
action of M0i, one may similarly show by using xi∗[···] = xi(e2Eκ ⊲[···]). One may similarly
demonstrate that the relation (13) is satisfied for other cases too. This implies that x(n+1)
satisfies Eq. (13).
We emphasize that the action of Ma is not equivalent to the action of xa(P ⊲) if the
b b
target is a tensor composed of whole module space since
κ
V
xa(P ⊲k xd) = iδdxak = Ma ⊲k xd. (15)
b c ∗ − b c 6 b c ∗
Eq. (13) holds only when the target space of (g) is the coordinates vector space which is
κ
U
a subset of .
κ
V
6
III. ACTION ON FUNCTIONS AND ⋆-PRODUCT BETWEEN FUNCTIONS
We now consider the action of igl(4,R) generators on the algebra of functions F =
F,m , where F denotes the space of functions and m denotes the ordinary product given
{ }
by m[f g](x) = [f(x) g(x)] = f(x)g(x), where f,g F. For convenience, we introduce
a new n⊗otation ◮·denoti⊗ng the action ⊲ of the generato∈rs on a function f with the form:
∂
Ma ◮ f(x) (Ma ⊲f)(x) = ixa f(x), (16)
b ≡ b − ∂xb
∂
P ◮ f(x) (P ⊲f)(x) = i f(x).
a a
≡ − ∂xa
Explicitly, thetwo actionsofMa onasimple functionk(x) = k xa F give different results,
b a ∈
Ma ⊲(k xa) = 0, Ma ◮ k(x) = ixak . (17)
b a b − b
In the first action ⊲, Ma acts on bothk and xa so that k xa is invariant under GL(4,R) and
b a a
in the second action ◮, it acts only on xa. On the other hand the two actions of momentum
are equivalent:
P ◮ f(x) = P ⊲f(x),
a a
since the action of momentum to the vector k vanishes.
a
Given the action ◮ of the algebras on F, we may define an associated algebra F = (F,⋆)
κ
as module algebra of (g) by using the twist (7). The ⋆-product between two functions
κ κ
f,g F is given by tUwisting the ordinary proFduct m by using the action ◮ as
κ
∈
f(x)⋆g(x) m [f g](x) := m −1 ⊲(f g) (x) (18)
κ κ
≡ ⊗ F ⊗
= (cid:2) −1 ◮ (f(x) (cid:3)g(x)) .
κ
· F ⊗
(cid:2) (cid:3)
If the functions f(x) and g(x) are ordinary functions without ⋆-product, Eq. (18) leads to
the conventional κ-Minkowski star-product,
i
f(x)⋆g(x) := exp D E E D ◮ f(x) g(x) (19)
·(cid:18) (cid:20)2κ ⊗ − ⊗ (cid:21) ⊗ (cid:19)
(cid:16) (cid:17)
i ∂ ∂ ∂ ∂
= exp yk xk (f(x) g(y)) .
(cid:18) (cid:20)2κ ∂x ∂y − ∂x ∂y (cid:21) · (cid:12)
(cid:16) 0 k k 0(cid:17) (cid:12)y→x
(cid:12)
(cid:12)
One may also try to get a deformed associated algebra by acting the action ⊲ on f(x)
rather than on its functional form f, as in Eq. (18). However in this case, the resulting
module algebra will be the same as the original one, F, since any action of generators Ma
b
on a scalar makes it vanish. Thus the star-product k xa defines a scalar function k,
a
∗
k(x) = k xa
a
∗
due to k xa = k xa and Eq. (17). Consider two scalar functions k(x) = k xa and
a a a
∗ ∗
q(x) = q xa so defined. Explicit calculation shows that the two functions commutes for
a
∗
the -product,
∗
k(x) q(x) = q(x) k(x) = k(x)q(x).
∗ ∗
7
Thisimpliesthatthe -productbetweenfunctionsreducestotheordinaryproduct. However,
∗
the ⋆-product does not commute,
i
k(x)q(x)+ (k q k q )xi = k(x)q(x). (20)
0 i i 0
2κ − 6
From Eqs. (11) and (20), one notices that the ⋆-product satisfies
k(x)⋆q(x) = k q (xa xb) = q(x)⋆k(x). (21)
a b
∗ 6
This is the reason why we construct the noncommutative star product between scalar func-
tions using ⋆.
Finally, the partial derivative ∂ is identified with iP and its coproduct is defined by
a a
∆ (∂ ) = i∆ (P ),
κ a κ a
∂ ◮ (φ(x)⋆ψ(x)) = (∂ ◮ φ(x))⋆ψ(x)+φ(x)⋆(∂ ◮ ψ(x)), (22)
0 0 0
∂i ◮ (φ(x)⋆ψ(x)) = (∂i ◮ φ(x))⋆ e2Eκψ(x) + e−2Eκφ(x) ⋆(∂i ◮ ψ(x)).
(cid:16) (cid:17) (cid:16) (cid:17)
We have shown in Ref. [31] that the differential structure is consistent with the Jacobi
identity and the relation d2 = 0.
κ
IV. ⋆-PRODUCT AND VECTORS
Up to this point, we have defined the ⋆-product between the scalar functions. However, it
is not yet defined the ⋆-product between a vector and a function. To find one, we calculate
the action of P on the product of two scalar functions in two different ways. First, we
a
calculate it by using the definition of coproduct (8),
P ◮ [k(x)⋆q(x)] = ik ⋆[eE/(2κ) ◮ q(x)] i[e−E/(2κ) ◮ k(x)]⋆q (23)
i i i
− −
1
= i[k ⋆q(x)+k(x)⋆q ]+ [k ⋆q k ⋆q ],
i i 0 i i 0
− 2κ −
P ◮ [k(x)⋆q(x)] = i[k ⋆q(x)+k(x)⋆q ].
0 0 0
−
Second, we calculate the ⋆-product before acting the momentum operator,
i
P ◮ [k(x)⋆q(x)] = P ◮ k(x)q(x)+ [k q(x) k(x)q ] (24)
a a 0 0
(cid:18) 2κ − (cid:19)
k q k q
0 a a 0
= i[k q(x)+k(x)q ]+ − .
a a
− 2κ
Since the two results should be the same we have the ⋆-product between k ’s and a function:
a
k ⋆q(x) = k q(x) = q(x)⋆k , k ⋆q = k q . (25)
a a a a b a b
This shows that k commutes with the ⋆-product so that
a
k(x)⋆q(x)⋆[P ◮ r(x)]⋆ ⋆s(x) = k(x)⋆q(x)⋆( ir )⋆ ⋆s(x)
a a
··· − ···
= ir [k(x)⋆q(x)⋆ ⋆s(x)].
a
− ···
8
The same result holds for the contravariant vector ka.
We can also do the same calculation for Ma as in Eqs. (23) and (24). For example, we
b
calculate
M0 ◮ [k(x)⋆q(x)]
i
k ⋆q x0 x0k ⋆q +xjk ⋆q k ⋆q xj
= i[k x0 ⋆q(x)+k(x)⋆x0q ]+ 0 i − i i j i − i j
i i
− 2κ
which should be the same as
1
M0 ◮ [k(x)⋆q(x)] = x0 i[k q(x)+k(x)q ]+ [k q k q ] .
i (cid:18)− i i 2κ 0 i − i 0 (cid:19)
Equating the two equations for all Ma, we get
b
i
xa ⋆k(x) = xak(x)+ (δak δak )xi, (26)
2κ 0 i − i 0
i
k(x)⋆xb = k(x)xb + k δb k δb xi.
2κ 0 i − i 0
(cid:0) (cid:1)
We now provide an independent check of Eqs. (25) and (26). One may conjecture that
k [xa ⋆f(x)] = (k xa)⋆f(x). However, one immediately realizes that this conjecture is
a a
∗ ∗
not valid since Eq. (23) is different from (24) with this conjecture. Noting k xa = k xa
a a
∗
and Eq. (21), one has to try a weaker form:
k [xa ⋆q(x)] = (k xa)⋆q(x) = k(x)⋆q(x). (27)
a a
∗
Then, by assuming xa ⋆q(x) = xaq(x)+ i Ba in Eq. (27), Eq. (20) results in Eq. (26). In
2κ
addition, one may have k ⋆f(x) = k f(x) as in Eq. (25) if one requires
a a
xa ⋆[k ⋆f(x)] = k(x)⋆f(x) = k ⋆[xa ⋆f(x)].
a a
Given the relations (25) and (26), we may show that the actions Ma and P satisfy
b a
Ma ◮ [k(x)⋆q(x)] = xa(P ◮ [k(x)⋆q(x)]).
b b
Similarly, we also get
Ma ◮ k1(x)⋆k2(x)⋆ ⋆kn(x) = xa P ◮ k1(x)⋆k2(x)⋆ ⋆kn(x) (28)
b ··· b ···
(cid:2) (cid:3) (cid:0) (cid:2) (cid:3)(cid:1)
which is consistent with Eq. (13).
In general, one may construct any function as a series of k(x). Therefore, Eq. (28) must
be satisfied for all scalar functions. In this sense, we propose the transformation law of a
scalar field f F under a general linear group as
κ
∈
(Ma ⊲f)(x) Ma ◮ f(x) = xa[P ◮ f(x)]. (29)
b ≡ b b
Especially, the time-symmetric exponential function
[eikx]s = eik02x0 ⋆ei~k·~x ⋆eik02x0, (30)
and their product [eikx] ⋆ [eiqx] satisfy the property (29). It is remarked in passing that
s s
other ordered exponential functions also satisfy the same property (29). Note that we have
used the ⋆-product to define the ordered exponential function in Eq. (30) rather than the
-product. This is a crucial difference from the previous work [31] in which the ordered
∗
product was implemented by using the -product and k was absent in the module algebra
a
∗
.
κ
V
9
V. ACTION OF A FREE SCALAR FIELD
In this section, we construct an action of a free scalar field which is invariant under the
action of general linear algebras in the sense of Eqs. (2) and (4).
To find a metric we choose k and q in Eq. (20) as (eµ) and (eν) , the tetrad of the
a a a b
coordinates (Note that the tetrad is independent of the coordinate vector xa because of
Eq. (4)). Multiplying the flat Minkowski metric η to Eq. (20) we arrive at
µν
η eµ(x)⋆eν(x) = η (eµ) (eν) xaxb = g xaxb. (31)
µν µν a b ab
g = η (eµ) (eν) with signature ( +++) transforms under the actions of igl(4,R) gen-
ab µν a b
−
erators as
Ma ⊲g = i(g δa +g δa), P ⊲g = 0.
b cd bd c cb d a cd
Since the RHS of Eq. (31) is invariant under the general linear transformation, the LHS
satisfies the transformation rule,
Ma ⊲η eµ(x)⋆eν(x) = 0. (32)
b µν
To construct a free scalar field theory in κ-Minkowski spacetime from twist, we need to
know some useful identities forthe time-symmetric exponential function (30). FromEq. (25)
we have
∂ ◮ [eiqx] = iq [eiqaxa] . (33)
a s a s
(This simple relation is also satisfied by the exponential function of different ordering if
one uses different twist parameter α). Eq. (33) confirms that the exponential function acts
as a scalar function following Eq. (29). In addition, the multiplication of two exponential
functions is given by a new scalar exponential function:
e
[eikx] ⋆[eiqx] = [ei(k+q)x] , (34)
s s s
where (k+q) = (k0 +q0,kie2qκ0 +qie−k2κ0). Note that k+q = q+k.
6
In the presence of a non-commutativity we introduce the integration of a scalar function
e e e
using the property,
d4x√ g ⋆φ(x) = d4x√ gφ(x) = √ g d4xφ(x).
Z − Z − − Z
In the first equality, we use Eq. (25) and in the second equality, we use the fact that the
metric tensor g is independent of coordinate vector xa. (If the metric were dependent on
ab
coordinate vector, the ⋆-product might be relevant in this calculation.)
We calculate the ⋆-product of two exponential functions to get a δ-function,
d4x√ g[e−ikx]s ⋆[eiqx]s = (2π)4√ ge32qκ0δ4(k q).
Z − − −
Since the left-hand-side is a scalar quantity, the right-hand-side is also a scalar under the
transforms in κ(g). Note that the δ-function appears with the normalization factor e32qκ0.
U
10
We define the Fourier transform of a scalar field with the form:
φ(x) = dµ [eikx] φ , (35)
k s k
Z
where dµ is an appropriate measure to be determined below. The inverse Fourier transform
k
must be given by the ⋆-product. We have the consistency condition,
φk = d4x√ g[e−ikx]s ⋆φ(x) = √ g dµqφq(2π)4e32qκ0 δ4(k q),
Z − − Z −
which determines the measure
1 d4qe32qκ0
dµ = . (36)
q
√ g (2π)4
−
The Hermitian conjugateof φ becomes φ†(x) = dµ [eikx] φ¯ ,where φ¯implies the complex
k s −k
¯
conjugate of φ. If φ(x) is a real scalar field, theRmode function satisfies φk = φ−k.
The integration of the product of two scalar fields is
d4x√ g ⋆φ(x)⋆ψ(x) = √ g d4xφ(x)⋆ψ(x) (37)
Z − − Z
= √1 g Z d(42kπd)48qe3(k02+κq0)φkψqZ d4x[ei(k+eq)x]s
−
= dµ φ ψ .
k k −k
Z
One may calculate any number of products of scalar fields in a similar way.
The action for a free scalar field in commutative spacetime with metric g is
ab
1
S = d4x√ g gab(∂ φ(x))(∂ φ(x)) m2φ(x)φ(x) . (38)
commutative a b
2 Z − − −
(cid:2) (cid:3)
This action is invariant under the general linear transformation in 4-dimensions in the sense
of linear diffeomorphism: Ma ⊲S = 0 and P ⊲S = 0. The corresponding action in noncom-
b a
mutative spacetime is obtained by the following modifications: 1) The partial derivative is
replaced by the nontrivial action ∂ ⊲ on ordered functions. 2) The normal products between
a
functions and vectors are replaced by the ⋆-products. Since the metric is independent of
coordinate vector, the ⋆-product between the metric and a function is irrelevant. Thus, the
action of a free real scalar field in κ-Minkowski spacetime from twist is written as
1
S = d4x√ g gab(∂ ◮ φ(x))⋆(∂ ◮ φ(x)) m2φ(x)⋆φ(x) . (39)
a b
2 Z − − −
(cid:2) (cid:3)
Note the way how P ◮ [eikx] transforms under the action Ma ◮:
a s b
Ma ◮ (P ◮ [eikx] ) = [Ma,P ] ◮ [eikx] +P ◮ (xa(P ◮ [eikx] )) = k k xa[eikx] .
b c s b c s c b s b c s
Therefore, P ◮ [eikx] transforms the same way as that of a scalar function under Ma.
c s b
By using Eq. (37), the action (39) can be expressed in terms of Fourier modes in a quite
simple form,
1
S = dµ φ (k2 +m2)φ , (40)
k k −k
−2 Z